On a boundary value problem for conically deformed thin elastic sheets

TL;DR

This paper investigates the elastic energy scaling of a thin, conically deformed elastic sheet with boundary constraints, using a modified bending energy penalization in the von Kármán model.

Contribution

It introduces new energy bounds for conically deformed sheets with non-quadratic bending energy penalization, extending the understanding of elastic behavior in thin sheets.

Findings

Established energy bounds scale as h^{p/(p-1)} for p in (2, 8/3)

Provided ansatz free upper and lower bounds for the elastic energy

Extended the analysis of elastic sheets beyond the classical quadratic bending energy

Abstract

We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. We define the free elastic energy as a variation of the von K\'arm\'an energy, that penalizes bending energy in $L^p$ with $p\in (2,\frac83)$ (instead of, as usual, $p=2$). We prove ansatz free upper and lower bounds for the elastic energy that scale like $h^{p/(p-1)}$, where $h$ is the thickness of the sheet.